Delete/unlink bones in rigify
Okay, so the issue is with your deform bones, particularly the facial deform bones. If you wanted to use them, you could re-align them, but since you do not, it is easier to just go ahead and delete them and then re-parent the armature. Here's how:

STEP 1

Select your armature and then under the Armature Properties panel, switch to Rest Position. Secondly, switch your active Armature Layer to be only the 3rd from the bottom most left. (See picture below, circled in red).



STEP 2

With the armature selected and only the deform layer active, tab into edit mode. Now you can delete all the bones you do not need. For this example, I selected the three head bones you probably will want (see selected bones in the photo below). Then I hid them (hotkey H) so that I would not accidentally delete those ones that I need. After they were hidden, I selected all the face deform bones and deleted them. 


NOTE: The selected bones in this photo are the ones you should NOT delete. You can delete the rest of the face bones for your purpose (the ones around the eyes, ears, and jaw)

STEP 3

After having deleted the bones you do not want. Go back to object mode and select your mesh. Then go to the Object Data property tab and under Vertex Groups, click the arrow button and select "Delete All Groups". Don't worry, one more step and we're almost done!



STEP 4

Now, with your mesh selected in object mode, select your armature so that it is the active object (orange outline instead of yellow). Then parent (control + P) the mesh to the armature using Automatic Weights. Go back to Pose Position and check out your animation. It might need some tweaking, but you will not have any facial issues when exporting down the road.

EDIT: I deleted some posts because I did not realize the format was to keep editing my first answer. I apologize, but wanted to post a comment that got deleted in the clean-up about the OP's FPX export settings. 
@Alberto Elorduy: By the way, my main fbx export settings are Selected only, Armature, Mesh, and empty. In armature only export deform bones, don't apply modifiers, and smoothing to faces. In animation I choose baked animation, key all bones and force start/end.